As an exhaust gas apparatus of an internal combustion engine to be used by an automotive vehicle and other vehicles, there is known an exhaust gas apparatus as shown in FIG. 32 (for example see Patent Document 1). In FIG. 32, the known exhaust gas apparatus 4 is constructed to allow an exhaust gas to be introduced therein after the exhaust gas exhausted from an engine 1 serving as an internal combustion engine passes through an exhaust manifold 2 and is then purified by a catalytic converter 3.
The exhaust gas apparatus 4 is constituted by a front pipe 5 connected to the catalytic converter 3, a center pipe 6 connected to the front pipe 5, a main muffler 7 connected to the center pipe 6 and serving as a sound deadening device, a tail pipe 8 connected to the main muffler 7, and a sub-muffler 9 connected to the tail pipe 8.
The main muffler 7 has an expansion chamber for introducing therein and expanding the exhaust gas to mute the sound of the exhaust gas, and a resonance chamber for muting the sound of the exhaust gas having a specified frequency under the influence of Helmholtz resonator effect. More specifically, the resonance chamber can tune its resonance frequency to the low frequency side by making large the volume of the resonance chamber or otherwise by making long the projecting length of the center pipe 6, while can tune its resonance frequency to the high frequency side by making small the volume of the resonance chamber or otherwise by making short the projecting length of the center pipe 6.
The sub-muffler 9 is adapted to reduce the sound pressure when the column air resonance is generated in the tail pipe 8 in response to the pipe length of the tail pipe 8 by the pulsation of the exhaust gas during the operation of the engine 1.
In general, the tail pipe having an upstream opening end and a downstream opening end at the respective upstream and downstream sides of the exhaustion direction of the exhaust gas is subjected to incident waves caused by being reflected with the pulsation of the exhaust gas during the operation of the engine at the upstream opening end and the downstream opening end, thereby causing an air column resonance with a wavelength. The air column resonance has a basic component of a frequency with a half wavelength equal to the pipe length of the tail pipe, and thus has a wavelength several times the half wavelength.
For example, taking an example in which the tail pipe 8 having no sub-muffler 9 extends backwardly from the main muffler 7, as shown in FIG. 33, the wavelength λ1 of the air column resonance of a basic vibration (primary component) is roughly double the pipe length L of the tail pipe 8, while the wavelength λ2 of the air column resonance of the secondary component is roughly one time the pipe length L of the tail pipe 8. Therefore, the tail pipe 8 has therein standing waves having respective node portions of sound pressures at the upstream opening end 8a and the downstream opening end 8b. 
The air column resonance frequency “fm” of the tail pipe 8 is given by the following equation (1).fm(c/2L)×m  (1)
c: sound speed, L: pipe length of tail pipe, m: degree
As will be obvious from the above equation (1), it is known that the longer the pipe length L of the tail pipe 8, the more the air column resonance frequency “fm” is transferred to the low frequency area with the rotation number of engine 1 being low.
It is further known that as shown in FIG. 34, the frequency of the exhaust gas pulsation of the engine 1 is increased as the rotation number of the engine 1 is increased, and the sound pressure levels (dB) of the exhaust gas sound are raised with the primary component f1 and the second component f2 of the exhaust gas sound caused by the air column resonance in response to the rotation number of the engine 1.
Therefore, in the case of using a tail pipe 8 having a long pipe length (for example, the pipe length of the tail pipe 8 is more than or equal to 1.5 m), there is occasionally generated such an air column resonance in the normal rotation area (2000 rp˜5000 rpm) having a low engine rotation number, thereby causing exhaust gas noises to be increased and thus giving unpleasant feelings to a driver.
In particular, as shown in FIG. 32, the peak (the width of the abdominal portion in the sound pressure distribution) of the sound pressure for the primary component f1 of the air column resonance is larger than the peak of the sound pressure for the secondary component f2 of the air column resonance, so that there is generated in the normal rotation area of the engine unpleasant noises called muffled sounds which are a cause for the exhaust gas noises to be deteriorated.
For this reason, in the case of the pipe length of the tail pipe 8 being long, the sub-muffler 9 smaller in capacity than the main muffler 7 is provided at the abdominal portion of the standing wave high in the sound pressure level, and at the optimum position among the respective abdominal portions of the primary component f1 and the secondary component f2 of the exhaust gas sound caused by the air column resonance as shown in FIG. 32, so that the exhaust gas noises can be suppressed in the normal rotation area of the engine to prevent the unpleasant feelings from be given to the driver.